As is known per se, a linear ARMA filter can be represented in the form of a transfer function H(z), in a field of z-transforms linking the input signal X(z) and the filtered output signal Y(z) by:
      Y    ⁡          (      z      )        =                    H        ⁡                  (          z          )                    ·              X        ⁡                  (          z          )                      =                                                      b              1                        +                                          b                2                            ·                              z                                  -                  1                                                      +            …            +                                          n                                  n                  +                  1                                            ·                              z                                  -                  n                                                                                        a              1                        +                                          a                2                            ·                              z                                  -                  1                                                      +            …            +                                          a                                  m                  +                  1                                            ·                              z                                  -                  m                                                                    ·                  X          ⁡                      (            z            )                              =                                    B            ⁡                          (              z              )                                            A            ⁡                          (              z              )                                      ·                              X            ⁡                          (              z              )                                .                    
The polynomial B(z) defines the moving average part (or Moving Average MA) of the ARMA filter, and the polynomial A(z) defines the autoregressive part (AR) of the ARMA filter.
Finite impulse filters (FIR) correspond in the event where the degree m of the polynomial A is zero.
Infinite impulse filters (IIR) correspond in the event where the degree m of the polynomial A is non-zero.
Because ARMA filters vary over time, and IIR filters in particular, they are used for example in sound-editing software for dynamically changing effect (multi-band equalisers, audio effects, etc.), in synthesis solutions (vocal and other) or in processing involving pickup or restoration of sound (reduction or cancellation of noises, whereof the characteristics vary over time).
Modification over time of ARMA filters, and more particularly of IIR filters, does however have disadvantages, mainly with the appearance of transitory artefacts during filter change. These artefacts manifest for example in the form of audio “clicks” particularly unpleasant to hear, and are for example associated with initialisation of internal memories of a new IIR filter to be activated, with dephasing introduced by the current ARMA filter to be switched off, and with potential differences in overall delay (group delay) of the above filters.
Hereinbelow, the description will concentrate mainly on the case of filters of IIR type, linear in particular. In fact, infinite impulse response filters are those for which a filter change conventionally generates the most artefacts. Also, linear filters prove to be much more widespread and studied than non-linear filters. In this way, the explanations hereinbelow could be applied directly in a wide range of classic situations.
Nevertheless, it is understood that the invention can apply to any type of ARMA filter, linear or non-linear, and especially in the case of Volterra filters which are non-linear polynomial filters. The theory of so-called polynomial “Volterra” filters is especially documented in the publication “The Volterra and Wiener theories of nonlinear Systems” (Schetzen, 1989, Ed. Krieger).
Different solutions have already been brought up in techniques of the prior art for trying to limit, or even cancel, the transitory effects linked to internal memories of these IIR filters.
One approach, illustrated by the publication “Elimination of transients in adaptive filters with application to speech coding” by Zetterberg, L. H. and Q. Zhang (1988), is based on initialisation of variables of internal state in internal memories of the new IIR filter to be activated.
For correct initialisation, a sufficient number of samples preceding the moment of IIR filter change is stored, then all these samples are filtered by the new IIR filter. Once initialised, the new IIR filter is applied to the signal.
The authors of this publication have shown that, contrary to theory, it is possible to be content with a reduced number of samples for executing this initialisation, despite the infinite character of the impulse responses of these filters. This number depends directly on the “length of the impulse response” of the new filter, a notion defined by the LONG first samples accounting for x % of the total energy of the filter (the percentage x being determined empirically and according to applications), in the case of a stable IIR filter whereof the impulse response tail tends to 0.
This approach however suffers from complexity and/or latency.
In fact, initialisation of internal variables of the new IIR filter requires filtering of a number of samples N which depends on the LONG length of the impulse response of the filter: in general, it is necessary for N>LONG to avoid any audible artefact.
Given this constraint, two options for execution are considered:                either the internal variables are initialised at the time of IIR filter change, requiring N+1 samples to be filtered: the N samples preceding the current sample and the current sample, though this generates a substantial complexity peak at the instant t of the IIR filter change;        or parallel filtering is carried out with the two filters: the switching between the two filters is done after N samples. This option involves latency of N samples before application of the new IIR filter, meaning that the new filter is applied at the end of these N samples only. The result is a significant hindrance if the aim is to apply the new IIR filter to the signal (for example for cancelling a particular signal) as fast as possible or again change IIR filter before the end of the N samples.        
These disadvantages are also there in the case of simple deactivation (switching off) of the current IIR filter (in this case, the “new filter” can be seen as an “identity filter” letting everything pass through), since the current IIR filter introduces dephasing in the filtered signal, the origin of audio “clicks” when the filter is switched off suddenly.
Finally, this approach fails to protect against differences in delay (group delay) between the two IIR filters: if the two filters have a different group delay, this inevitably becomes an audible artefact (‘click’ type) linked to discontinuity of the signal.
A hybrid approach has also been covered in the publication “A modified-superposition speech synthesizer and its applications” by Verhelst, W. and P. Nilens (1986).
This approach uses a “vanishing technique” consisting of instantiating the two IIR filters in parallel at the time of switching from one filter to the other, with the following configuration: the new IIR filter, non-initialised, receives the signal to be filtered whereas the former IIR filter “switches off” (by vanishing) and receives no input.
The filtered output is constituted by the sum of the outputs of each of the two IIR filters, a sum formed during a limited period N. In practice, the “interpolation” time N (the duration of mixing of the outputs) necessary for having no audio artefact is greater than the LONG lengths of the impulse responses of the two IIR filters.
This hybrid approach relates to the second execution option mentioned hereinabove for the approach with initialisation of internal memories. In fact, use is also made of two IIR filters operating in parallel.
The difference however is in the instantaneous transition between the two IIR filters, made possible due to mixing the two outputs (rather than switching instantaneously from an output of one filter to the other after a certain time in the second option hereinabove). This limits the artefacts linked to differences in group delay between the IIR filters.
However, this hybrid approach also suffers from other disadvantages mentioned previously, and especially from latency due to the interpolation duration which must remain greater than the LONG effective length of the impulse response of the IIR filters.